[[1]}] 
Let Assumptions REF , REF  and Assumptions REF , REF , REF  in Appendix  hold. Then, there exists at least one solution to the functional equation

Y|A=a, Z=z = h(a, W) W(w|A=a, Z=z) w, 

which holds for any \((a,z) \in \mathcal {A} \times \mathcal {Z}\) .
Here, we denote \(\rho _W(w|A=a, Z=z)\)  as the density function of the conditional probability \({W|A=a, Z=z}\) . Let \(h^*\)  be a solution of (REF ). The structural function \(f_{\mathrm {struct}}\)  is given as

fstruct(a) = [W]h*(a,W) .


For completeness, we present a proof in Appendix  (Lemma REF  and Corollary REF ), which is due to [2]} and [3]}. Following [2]}, we call \(h^*\)  the bridge function. From Proposition REF , we can see that the estimation of the structural function reduces to the estimation of the bridge function, since once we obtain the bridge function, the structural function directly follows from (REF ).
